# Inducing currents without change of flux linkage?

• I

## Main Question or Discussion Point

Suppose I had cylindrically-symmetric rotating magnet surrounded by a plasma.

I rotate it on its axis at a constant angular velocity, and so the electric field E produced is non-solenoidal and can be described as the negative gradient of some potential V(x,y,z).

The electric field is induced perpendicularly to the vector potential, as the vector potential points in the azimuthal (φ) direction and the electric field has no azimuthal (φ) components.

So some current is expected to develop at right angles to the vector potential.

However, when current develops at right angles to the vector potential of some source, that current does not induce net change of flux linkage directed on the magnet. Or does it?

Since the magnet is not one monolithic object, the subobjects (magnetic domains) which make it up have different velocities. In the frame of reference of one of these subobjects (magnetic domains), the external magnetic field B from the plasma is transformed to a magnetic field B' observed in its instantaneous co-moving inertial frame. This suggests that in the instantaneous co-moving inertial frame of a magnetic domain, the electric field E' may have solenoidal components which are absent in the lab frame.

Since the curl of the magnetization M gives us a magnetization current, analogous to an electric current, the emf induced on these "magnetization currents" through the closed-line integral of the electric field, corresponds to energy exchange through electromagnetic induction. Thus, while the lab frame could suggest that no EMF is occurring on these magnetization currents (i.e. curl E = 0), in the local "material" frame of each magnetic domain, there is a curl E' due to an apparent changing magnetic field.

Also, as each magnetic domain revolves around the collective axis, the transformation of the apparent E' and B' fields change with time. The time derivative of B' would differ in the material frame from the lab B field by the time derivative of - v x E / c^2 (in the non-relativistic approximation). I would assume that the changes of velocity v of a magnetic domain with time, which leads to apparent changes in the externally applied magnetic field B'(v), does not generate a physical EMF on its magnetization currents, but that the apparent change of EMF due to changes in external field B'(E(t,r(t))) observed in the instantaneous co-moving inertial frame (which is offset from the lab frame by instantaneous velocity v), ultimately due to time and spatial variations of E in the lab frame, would result in a physical EMF. If this is true, wouldn't that mean the time integral of the physical EMF on the magnetization currents could accumulate (volt-seconds in SI units) even though B is constant (i.e. curl E = 0)? To put this in another way, is it right to think that a changing magnetic field B' would be observed for a looped magnetization current traveling at velocity v perpendicular to an electric displacement current ∂E/∂t, which would be subject to an EMF (closed loop integral of E'(t,r(t))) that is not apparent in the lab frame?

Would then the work done on the plasma through the potential V(x,y,z) produced by a cylindrically-symmetric magnet with constant angular velocity be possible by extracting energy from the magnetization currents of a magnetic domain through the volume integral of the time integral of curl M dot E', disregarding those changes of E' which are specifically due to time variation of the velocity v of a magnetic domain?

Related Other Physics Topics News on Phys.org
anorlunda
Mentor
I rotate it on its axis
That's ambiguous. You can draw 3 axes through the magnet.

This whole thing sounds like an exercise in Maxwell's Equations. Is that what you are looking for, a page full of math? a textbook citation?

No matter what type of answer, it should start with a sketch that shows the exact meaning for any phrase such as "the angle." Without a sketch, miscommunication is very likely. Use the UPLOAD button to insert pictures into your post.

• sophiecentaur
That's ambiguous. You can draw 3 axes through the magnet.

This whole thing sounds like an exercise in Maxwell's Equations. Is that what you are looking for, a page full of math? a textbook citation?

No matter what type of answer, it should start with a sketch that shows the exact meaning for any phrase such as "the angle." Without a sketch, miscommunication is very likely. Use the UPLOAD button to insert pictures into your post.
Sorry for being unclear. However, I believe that the answer was implied by the statement:

I rotate it on its axis at a constant angular velocity, and so the electric field E produced is non-solenoidal and can be described as the negative gradient of some potential V(x,y,z).
So, I believe this implies that the magnetic field density B is not changing, which in turns implies that if the cylindrical magnet is rotating, it must be doing so on its axis of symmetry. It took me a few hours to write the post. Making pictures may take several more!

Fortunately I found a relevant picture. Fig 1b below may help: #### Attachments

• 3.3 KB Views: 216
Here is a paper that may help explain what I mean by "V(x,y,z)"

"Electromagnetic interactions derived from potentials: charge and magnetic dipole" by Roberto Coïsson
https://arxiv.org/pdf/1403.0973.pdf

However, he uses the symbol φ for the electric scalar potential rather than the azimuthal angle. The paper provides information on the electric scalar potential for a moving magnetic dipole. If I had a magnet rotating on its axis, I would figure that each volume differential element of the magnet could be ascribed a velocity and each element would possess a magnetization dm which contributes to the electric scalar potential according to equation 4 of Coïsson's paper.

For simplicity we could assume the Coulomb gauge, where the negative time derivative of the vector potential is responsible for the solenoidal component of the electric field, while the negative spatial derivative of the electric scalar potential is responsible for the irrotational component of the electric field, such that V(x,y,z) is essentially the same as the electric scalar potential. This "artificial split" is elaborately described by Kirk T McDonald in his paper "The Helmholtz Decomposition and the Coulomb Gauge" which can be seen at http://physics.princeton.edu/~mcdonald/examples/helmholtz.pdf

Kirk T. McDonald said:
Thus, the Helmholtz decomposition (1) and (5) of the electric field E is equivalent to the decomposition (3) in terms of a scalar and a vector potential, provided those potentials are calculated in the Coulomb gauge.
In contrast, Coïsson in his paper appears to be using the Lorenz gauge definition for the vector potential as implied by equation 3. Perhaps I should additionally specify that the electric scalar potential contributed by the rotating magnet is assumed to be constant, such its magnetic vector potential has no sources or sinks in the Lorenz gauge, allowing us to relate Coïsson's equation 3 to our problem where we are interested in the irrotational component of the electric field of a cylindrical magnet rotating on its sole axis of symmetry.

Edit: Come to think if it, it would be probably much simpler to calculate the effective polarization P = v x M of each differential volume element and determine V(x,y,z) that way.

https://physics.stackexchange.com/a/74830

Last edited: